Invitation to linear algebra

I: Matrices and Linear SystemsLesson 1 Introduction to Matrices Lesson 2 Matrix Multiplication Lesson 3 Additional Topics in Matrix Algebra Lesson 4 Introduction to Linear Systems Lesson

Invitation to Linear Algebra

TEXTBOOKS in MATHEMATICS

Series Editors: Al Boggess and Ken Rosen

PUBLISHED TITLES

ABSTRACT ALGEBRA: A GENTLE INTRODUCTION

Gary L Mullen and James A Sellers

ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH, SECOND EDITION

William Paulsen

ABSTRACT ALGEBRA: AN INQUIRY-BASED APPROACH

Jonathan K Hodge, Steven Schlicker, and Ted Sundstrom

ADVANCED LINEAR ALGEBRA

APPLIED ABSTRACT ALGEBRA WITH MAPLE™ AND MATLAB®, THIRD EDITION

Richard Klima, Neil Sigmon, and Ernest Stitzinger

APPLIED DIFFERENTIAL EQUATIONS: THE PRIMARY COURSE

Vladimir Dobrushkin

A BRIDGE TO HIGHER MATHEMATICS

Valentin Deaconu and Donald C Pfaff

COMPUTATIONAL MATHEMATICS: MODELS, METHODS, AND ANALYSIS WITH MATLAB® AND MPI, SECOND EDITION

Robert E White

A COURSE IN DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS, SECOND EDITION Stephen A Wirkus, Randall J Swift, and Ryan Szypowski

A COURSE IN ORDINARY DIFFERENTIAL EQUATIONS, SECOND EDITION

Stephen A Wirkus and Randall J Swift

DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE, SECOND EDITION

Steven G Krantz

DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE WITH BOUNDARY VALUE PROBLEMSSteven G Krantz

DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES, THIRD EDITION

George F Simmons

DIFFERENTIAL EQUATIONS WITH MATLAB®: EXPLORATION, APPLICATIONS, AND THEORY

Mark A McKibben and Micah D Webster

DISCOVERING GROUP THEORY: A TRANSITION TO ADVANCED MATHEMATICS

Tony Barnard and Hugh Neill

DISCRETE MATHEMATICS, SECOND EDITION

Kevin Ferland

ELEMENTARY NUMBER THEORY

James S Kraft and Lawrence C Washington

EXPLORING CALCULUS: LABS AND PROJECTS WITH MATHEMATICA®

Crista Arangala and Karen A Yokley

EXPLORING GEOMETRY, SECOND EDITION

GRAPHS & DIGRAPHS, SIXTH EDITION

Gary Chartrand, Linda Lesniak, and Ping Zhang

INTRODUCTION TO ABSTRACT ALGEBRA, SECOND EDITION

INTRODUCTION TO NUMBER THEORY, SECOND EDITION

Marty Erickson, Anthony Vazzana, and David Garth

LINEAR ALGEBRA, GEOMETRY AND TRANSFORMATION

Bruce Solomon

PUBLISHED TITLES CONTINUED

MATHEMATICAL MODELLING WITH CASE STUDIES: USING MAPLE™ AND MATLAB®, THIRD EDITION

B Barnes and G R Fulford

MATHEMATICS IN GAMES, SPORTS, AND GAMBLING–THE GAMES PEOPLE PLAY, SECOND EDITION Ronald J Gould

THE MATHEMATICS OF GAMES: AN INTRODUCTION TO PROBABILITY

MEASURE THEORY AND FINE PROPERTIES OF FUNCTIONS, REVISED EDITION

Lawrence C Evans and Ronald F Gariepy

NUMERICAL ANALYSIS FOR ENGINEERS: METHODS AND APPLICATIONS, SECOND EDITION

Bilal Ayyub and Richard H McCuen

ORDINARY DIFFERENTIAL EQUATIONS: AN INTRODUCTION TO THE FUNDAMENTALS

TRANSFORMATIONAL PLANE GEOMETRY

Ronald N Umble and Zhigang Han

TEXTBOOKS in MATHEMATICS

Invitation to Linear Algebra

David C MelloJohnson & Wales UniversityProvidence, Rhode Island, USA

CRC Press

Taylor & Francis Group

6000 Broken Sound Parkway NW, Suite 300

Boca Raton, FL 33487-2742

© 2017 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S Government works

Printed on acid-free paper

Version Date: 20161121

International Standard Book Number-13: 9781498779562 (Hardback)

This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.

Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.

For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and

are used only for identification and explanation without intent to infringe.

Library of Congress Cataloging-in-Publication Data

Names: Mello, David C (David Cabral), 1950-

Title: Invitation to linear algebra / David C Mello

Description: Boca Raton : CRC Press, 2017 |

Includes bibliographical references and index

Identifiers: LCCN 2016053247 | ISBN 9781498779562 (978-1-4987-7956-2

Subjects: LCSH: Algebras, Linear—Textbooks

Classification: LCC QA184.2 M45 2017 | DDC 512/.5 dc23

LC record available at https://lccn.loc.gov/2016053247

Visit the Taylor & Francis Web site at

http://www.taylorandfrancis.com

and the CRC Press Web site at

http://www.crcpress.com

Unit II: Determinants

Unit IV: More About Vector Spaces

Unit VI: Matrix Diagonalization

viii

Unit VII: Complex Vector Spaces

Unit VIII: Advanced Topics

Unit IX: Applications

To the Instructor

This book is intended as an introductory course in linear algebra for

sophomore or junior students majoring in mathematics, computer science,economics, and the physical sciences Lofty assumptions have not been madeabout the level of mathematical preparation of the typical student; in fact, it isonly assumed that the typical student has completed a standard calculus

course, and is familiar with basic integration techniques

As a fellow instructor, I know that this course is probably the typical student’sfirst encounter with the requirement of formulating mathematical proofs, anddealing with mathematical formalism For this reason, each definition has beencarefully stated, and detailed proofs of the key theorems have been provided.More importantly, in each proof, the motivation for each step has been given,along with the “intermediate steps” that are normally omitted in most texts

Unlike most books of this type, the book has been organized into “lessons”rather than chapters This has been done to limit the size of the mathematicalmorsels that must be digested by your students during each class, and to make

it easier for you, the instructor, to budget class time Most lessons can becovered in a standard class period

Considerably more material has been provided than is normally covered

in a first course For example, several advanced topics such as Jordan

canonical form, and matrix power series have been included This is to

make the book more flexible, and allow you to choose enrichment materialwhich may reflect your interest and that of your students

In addition, numerous applications of the course material to mathematicsand to science appear in the exercises The special applications, consisting

of the application of linear algebra to both linear and nonlinear dynamicalsystems appear inLessons 36-38at the end of the text

I would like to thank my colleague Dr Adam Hartman, for carefully reviewingthe manuscript, and for his many helpful comments If you should have anyideas or suggestions for improving the book, please feel free to email me

directly at dmello@jwu.edu.

David C Mello

Providence, Rhode Island

October, 2016

To the Student

This book has been written with you, the student, in mind It has been designed

to help you learn the key elements of linear algebra in an enjoyable fashion.Hopefully, it will give you a glimpse of the intrinsic beauty of this subject, andhow it can be used in your chosen field of study

For most students such as yourself, linear algebra is probably your first

encounter with formal mathematics and in constructing proofs of mathematicalpropositions In using the book, you should pay close attention to the

definitions of new concepts, and to the various theorems; these items havebeen clearly designated throughout the entire text

Whenever possible, you should try to work through the proof of each

theorem Pay attention to each step, and make sure it makes sense to

you, before proceeding to the next step If you do this, you will be

rewarded with a deeper understanding of the course material, and it willmake the exercises involving proofs much easier

I often tell my students that learning mathematics is like learning to play

a musical instrument Listening to someone else play a beautiful melodymight be enjoyable, but it doesn’t help you master the instrument yourself

So, you have to take the time to actually “do mathematics” in order to

play its music

But exactly how do you “do mathematics?” Here are some helpful hints forlearning and doing mathematics:

1 Always read the text with a pencil in hand, and take the time to work

through each example provided in the text Numerous examples have beenprovided to help you master the course material

2 Before attempting to solve a problem that involves constructing your

own proof, look at the relevant definitions of the concepts involved so

you can clearly understand exactly what has to be demonstrated

3 Be sure to do the homework problems that are assigned by your

instructor If you are unable to do some of the assigned work, or if you

are confused about some point, then don’t get discouraged, but be sure

to ask your instructor for help

4 Prepare for each class by reading the assigned material in the text

before class If you take the time to do this, you’ll find that you will spend

xii

less time taking notes, and more time really understanding and enjoyingeach class.

I sincerely hope that you will find the textbook to be extremely helpful andthat you will enjoy “doing mathematics.” Don’t be afraid of new concepts

if you don’t fully understand them at your first reading; a great physicistonce said that “if you give mathematics a chance, and if you learn to lovemathematics, then it will love you back!”

I: Matrices and Linear Systems

Lesson 1 Introduction to Matrices

Lesson 2 Matrix Multiplication

Lesson 3 Additional Topics in Matrix Algebra

Lesson 4 Introduction to Linear Systems

Lesson 5 The Inverse of a Matrix

Arthur Cayley (1821-1895)

(Portrait in London by Barraud and Jerrard)

It is interesting to note that the general rules of matrix algebra were first

elucidated by the English mathematician Arthur Cayley, Sadlerian professor

of mathematics at Cambridge University In an important paper, published in

1855, Cayley formally defined the concept of a matrix, introduced the rules ofmatrix algebra, and examined many of the important properties of matrices

Although the basic properties of matrices were known prior to Cayley’s

work, he is generally acknowledged as the creator of the theory of matricesbecause he was the first mathematician to treat matrices as distinct mathematicalobjects in their own right, and to elucidate the formal rules of matrix algebra.Cayley received many honors for his mathematical work and made major

contributions to the theory of determinants, linear transformations, the

analytic geometry of n dimensional spaces, and the theory of invariants.

1 Introduction to Matrices

In this lesson, we’ll learn about a brand new type of mathematical object

called a matrix As we will see in the lessons that follow, matrices are very

useful in many different applications, and they are particularly useful in helping

us to solve systems of linear equations

A matrix is a rectangular array of objects that are called elements of the

given matrix We usually use a capital letter to denote any given matrix, and

we enclose its elements with two brackets Take a look at the first example

Example-1: The following rectangular arrays are matrices:

We say that a given matrix is an r × c matrix [read “r by c”], or has the size

or dimension r × c, if that matrix has exactly r rows and c columns In the

example above, we see that A is a 2 × 2 matrix, B is a 2 × 1 matrix, C is a 2 × 3 matrix, D is a 3 × 1 matrix, E is a 3 × 3 matrix, and F is a 1 × 3 matrix Observe

that when we specify the size or dimensions of a matrix, we always specify the number of rows first, and then we specify the number of columns.

A matrix is said to be a square matrix if it has the same number of rows and

columns The adjective “square” comes from the fact that a square matrix has

a square shape In the example above, it is easy to see that both A and E are

square matrices

A matrix is sometimes called a column vector if it just has one column, and

a row vector if it has only one row In the above example, we see that the

matrices B and D are column vectors, while the matrix F is a row vector.

In the discussion that follows, we will see that matrices have an algebra oftheir own, and in many ways this algebra resembles the algebra of real numbers

We shall first define what it means for two matrices to be equal

Definition: (Matrix Equality)

Let A and B both be m × n matrices, then we say that “A equals B” and agree

to write A = B if and only if each element of A is equal to each corresponding element of B.

Thus, for two matrices to be equal two things must happen: (1) the matrices

must have the same size, and (2) the corresponding elements of the two

matrices must be equal to each other Take a look at the next example

Example-2: Given that:

2x 3y

4 25

solve for x, y, z and w.

Solution: Since the two matrices are equal, then their corresponding

elements must be equal Consequently, we have:

2x = 4, 3y = 9, 4z = 4, 5w = 25

We conclude that x = 2, y = 3, z = 1 and w = 5.

We now turn our attention to the addition and subtraction of matrices, and theoperation of scalar multiplication, i.e., multiplying a matrix by a number (orscalar)

Definition: (Matrix Addition)

Let A and B both be m × n matrices Then their sum, denoted by A + B, is the

m × n matrix obtained by adding the corresponding elements of A and B.

Observe that we can add any two given matrices only when they have the same

size; otherwise, we say that their sum is not defined.

Example-3: Perform the following matrix addition:

1 2

−3 −4

Solution: Since both matrices have the same size, their sum is defined.

Now, we just add their corresponding elements:

4

Since each element of our answer is zero, we call this a zero matrix In general,

a zero matrix is any matrix whose elements are all zero.

Definition: (Matrix Subtraction)

Let A and B both be m × n matrices Then their difference, denoted by A − B,

is the m × n matrix which is obtained by subtracting the corresponding elements

of A and B.

In the above definition, we see that the subtraction of any two matrices only

makes sense when both matrices have the same dimensions; otherwise, we

say that their difference is not defined In this regard, matrix subtraction issimilar to the addition of matrices

Example-4: Perform the following matrix subtraction:

1 2

−3 −4

Solution: Since both matrices have the same dimensions, their difference

is defined We now subtract their corresponding elements:

Definition: (Scalar Multiplication)

Let A be an m × n matrix, and let c be any number - either real or complex Then the matrix cA is the m × n matrix obtained by multiplying each element

of A by the number c.

Example-5: Given that:

−1 −2

−3 5Evaluate each of the following expressions:

to perform the indicated operations If an operation is not defined, then explain why.

Find the matrix X.

In Exercises 24-26, use the 2 × 2 matrices:

3 6

1 9

to verify each of the following statements:

24 Matrix addition is commutative, i.e.,

A + B = B + A

8

25 Scalar multiplication is distributive:

c(A + B) = cA + cB for any scalar c

26 If the additive inverse of A is the matrix defined by:

− A = −1 ⋅ A then the matrix −A has the property that:

A + (−A) = (−A) + A = 0

where 0 is denotes the 2 × 2 zero matrix

27 An unknown 2 × 2 matrix X satisfies the equation:

2 Matrix Multiplication

In this lesson, we shall discuss the basis aspects of matrix multiplication It

will be convenient to first introduce a useful notational convention, which is

called double subscript notation We can use this shorthand notation to

represent any element of a given matrix

Definition: (Double Subscript Notation)

For any m × n matrix A, the shorthand notation:

A = [a ij] for i = 1, 2, 3, , m

j = 1, 2, 3, , n (2.1)

shall mean that the matrix A consists of the elements a ij , where the first subscript

i denotes the row of any given element, and the second subscript j denotes its

column In general, we can write:

Before we define matrix multiplication in a general way, we shall first define

the dot product of a row vector and a column vector The result of this new

operation is always a scalar; that is, a real or complex number

Definition: (Dot Product)

Let r be a 1 × p row vector:

then the dot product of r and c, denoted by r⋅ c, is the scalar

(number) given by:

r⋅ c = r1c1+ r2c2+ r3c3+ +r p c p (2.3)

In other words, to obtain the dot product r ⋅ c, we multiply each element

of the row vector r by the corresponding element of the column vector c

and add all of the resulting products

Example-3: Calculate the dot product:

−123

Solution: Using the above definition, we multiply each element of the row

vector by the corresponding element of the column vector, and then add the

resulting products:

−123

= (2)(−1) + (5)(2) + (4)(3) = 20

It turns out that two general matrices A and B can be multiplied together under

certain conditions If the number of columns in A is equal to the number of

rows in B, then we say that the product AB is defined.

12

Definition: (Matrix Multiplication)

Let A be an m × p matrix, and B be a p × n matrix Furthermore, let row i (A)

denote the i-th row of A, and col j (B) denote the j-th column of B Then the

product of A and B, denoted by AB, is the m × n matrix C = [c ij] whose

typical element c ij is obtained by taking the dot product of the i-th row of A,

and the j-th column of B Symbolically, we can write:

c ij = rowi (A)⋅ colj (B) where i = 1, 2, 3, , m

Solution:

(a) Here we have a 2 × 2 matrix times a 2 × 2 matrix According to the

previous definition, the size of the product will be a 2 × 2 matrix as well

Also, according to this definition, we must multiply each row of the first

matrix by each column of the second matrix:

(b) Once again, we have a 2× 2 matrix times a 2 × 2 matrix so the size

of the product will be a 2 × 2 matrix as well Once again, we multiply

each row of the first matrix by each column of the second matrix to get:

Observe that this answer was different from the one we got in (a) This means

that matrix multiplication is not commutative; in general, the order in

which we multiply any two matrices matters.

(c) Here we have a 2 × 2 matrix times a 2 × 1 matrix Since the number of

columns in the first matrix equals the number of rows in the second matrix,

then their product is defined According to the above definition, the size of

their product will be a 2 × 1 matrix We obtain:

Definition: (Kronecker Delta and Identity Matrix)

The Kronecker delta δ ij of order n, is defined by

δ ij = 1, if i = j

0, if i ≠ j for i, j = 1, 2, 3, , n (2.6)

An identity matrix I of order n is a square n × n matrix such that I = [δ ij]

In other words, a square matrix is said to be an identity matrix if all of its

elements on the main diagonal (going from the top left corner to the bottom

right corner) are all equal to one and all of its remaining elements are equal

Identity matrices possess a very important property: when any square matrix

is multiplied by the identity matrix of the same size, the product is always the

matrix you started with This idea is expressed in the following theorem:

14

Theorem 2.1: (Multiplication by an Identity Matrix)

Let A be an n × n matrix, and let I be the n × n identity matrix Then,

That is, I is the identity element for matrix multiplication, and multiplying

any given square matrix by I is commutative.

Proof: Let A = [a ij ], and I = [δ ij ] Furthermore, let [AI] ijdenote a typical

element of the product AI Clearly,

only when k = j Similarly,

Before we conclude this lesson, we summarize the basic rules of matrix

algebra for easy reference

Theorem 2.2: (Rules of Matrix Algebra)

If the sizes of the matrices A, B, and C are such that the indicated operations

are defined, then the following basic rules of matrix algebra are valid:

A + B = B + A (Commutative Law for Addition)

A(B + C) = AB + AC (Left Distributive Law)

c(A + B) = cA + cB for any scalar c.

Proof: We prove only the left distributive law Let A be an m × p matrix, and

assume that both B and C are p × n matrices Using the definitions of matrix

addition and multiplication, we can write

remaining properties are easy, and are left as exercises □.

Roughly speaking, the previous theorem says that matrices almost obey thesame laws as real numbers; however, there are two important exceptions:

(1) Matrix multiplication is not commutative in general,

(2) The zero factor property of real numbers is not satisfied; that is, if

we have matrices such that AB = 0, we cannot safely conclude that either

21 Use the matrices A, B, and C (above) to verify that matrix

multiplication is associative; that is, show that:

A(BC) = (AB)C

22 Use the matrices A, B, and C (above) to verify that matrix

multiplication distributes over matrix addition; that is,

A(B + C) = AB + AC

23 If A is any square matrix, then use the rules of matrix algebra to

show that:

(A + I)(A − I) = A2− I

24 If A and B are any n × n matrices, then use the rules of matrix

algebra to show that:

(A + B)2 = A2+ AB + BA + B2

Why can’t we just combine the terms AB and BA?

25 The transpose of an m × n matrix A, denoted by A T , is the n × m matrix whose columns are obtained from the rows of A For example, if

27 (Pauli Spin Matrices) An important collection of square matrices that

occur in the non-relativistic theory of electron spin is called the Pauli spin matrices These matrices are defined as:

3 Additional Topics in Matrix Algebra

In this lession, we shall explore some additional topics in matrix algebra whichprove useful in science and engineering Also, we shall examine some new types

of special matrices, and additional matrix operations

Definition: (Transpose)

Let A be an m × n matrix; then the transpose of A, denoted by A T , is the n × m matrix whose rows are formed from the successive columns of A In other words,

if A = [a ij ] and A T = [b ij ] is the transpose of A, then b ij = a ji

Note that if B = A T , then we form the rows of B by using the columns of A.

A simple example should clarify this idea

Example-1: Given the matrices A, B, and C, where

4

−32,

we find that their transposes are:

103,

T

From the above example, we see that the transpose of a row vector is a

column vector, and conversely, the transpose of a column vector is always arow vector The transpose obeys several important properties that we

summarize in the following theorem

Theorem 3.1: (Properties of Matrix Transposition)

If the sizes of the matrices A and B are such that the indicated operations are

defined, then the following basic rules of matrix transposition are valid:

(A T)T = A (A + B) T = A T + B T

(cA) T = cA T for any scalar c (AB) T = B T A T

(3.1a) (3.1b) (3.1c) (3.1d)

Proof: We only prove (3.1d) Assume that A is an m × p matrix, B is a p × n

matrix, and let C = AB Furthermore, let A T = a ij′ , B T = b ij′ , and

C T = c ij′ Finally, let [B T A T]ij denote a typical element of B T A T

so that B T A T = (AB) T as claimed The proofs of the remaining properties are

left as exercises for the reader □

There are special types of square matrices, called symmetric matrices and

anti-symmetric matrices that often appear in the application of matrices to

important problems in physics and engineering We define them as follows:

Definition: (Symmetric and Anti-Symmetric Matrices)

A square matrix A is symmetric if

A T = A,

while it is said to be anti-symmetric if

A T = −A

We note, in passing, that if a square matrix A = [a ij] is anti-symmetric, then in

particular, we must have a ii = −a ii so that all of its diagonal elements must be

zero On the other hand, a square matrix is symmetric if the elements of the

given matrix are symmetric with respect to the main diagonal

Example-2: Consider the matrices

20

It turns out that every square matrix has a split personality in the sense that it

can be written as the sum of a symmetric matrix and an anti-symmetric matrix

This is the content of the next theorem

Theorem 3.2: (Decomposition of a Square Matrix)

Every square matrix A can be written as the sum of a symmetric matrix A sym

and an anti-symmetric matrix A anti; that is,

so A sym and A antiare symmetric and anti-symmetric matrices, respectively

Finally, if we add these two matrices together, we obtain

Upon completing this lesson, we introduce one last important matrix operation,

called computing the trace of a square matrix The trace of any square matrix is

always a scalar (number), and often appears in the application of matrices to

physics and in the application of group theory to physical problems

Definition: (Trace of a Matrix)

Given a square matrix A of order n, the trace of A, denoted by Tr(A), is defined

as the sum of the diagonal elements of A; that is,

Tr(A) = ∑

i=1

n

So, according to (3.3), if we want to find the trace of any square matrix, then

all we have to do is add all of its elements along the main diagonal The trace

of a square matrix has several useful properties that are given in the last

theorem of this lesson

Theorem 3.3: (Properties of the Trace)

Let A and B be square matrices of the same size, then

Tr(A + B) = Tr(A) + Tr(B) Tr(cA) = cTr(A) for any scalar c Tr(AB) = Tr(BA)

Tr(A) = Tr(A T)

(3.4a) (3.4b) (3.4c) (3.4d)

Proof: The proofs of (3.4a), (3.4b) and (3.4d) are easy, and are left as

exercises for the reader Let’s prove (3.4c) To this end, let [AB] ij denote

a typical element of AB, and observe that

Note that in the last step, we interchanged the order of summation The result

follows upon comparing the right hand sides of the above equations □

22

σ0 = 1 0

0 −1are all symmetric

write A as the sum of a symmetric and anti-symmetric matrix.

14 Provide proofs of properties (3.1a), (3.1b), and (3.1c).

15 A square matrix D = [d ij ] is said to be a diagonal matrix if d ij = 0 for

all i ≠ j If D is a diagonal matrix of size n, show that for any natural number k,